Fix any \(p>1\)  and \(\frac{1}{2}<k_0<\frac{p}{2}\) . Let \(\frac{1}{2}<k_1<\min \lbrace 1,k_0\rbrace \)  and \(k\ge k_0\) . Let \(\alpha =2\)  when \(n\ge 3\) , and let \(\alpha \in (1,2)\)  when \(n=2\) . Choose \(q\in (2,\frac{n\alpha }{n-\alpha })\) . Set \(s_0=\frac{2q}{q+2}\) . Note that \(1<s_0<\frac{n}{n-2}\)  when \(n\ge 3\)  and \(1<s_0<\alpha \)  when \(n=2\) .
Define as in [1]} Lemma 3.4, for any \(\delta >0\) , \(N>>1\)  and \(\beta \ge k_0\) ,
\(H_{\delta ,N}(t)=\left\lbrace \begin{array}{l l}t^\beta ,&\quad t\in [\delta ,N],\\N^\beta +\frac{\beta }{k_1} N^{\beta -k_1}(t^{k_1}-N^{k_1}), &\quad t>N.\end{array}\right.\) 
Then
\(H^{\prime }_{\delta ,N}(t)=\left\lbrace \begin{array}{rl}\beta t^{\beta -1},&\quad t\in (\delta ,N),\\\beta N^{\beta -k_1}t^{k_1-1}, &\quad t>N.\end{array}\right.\) 
Define, furthermore,
\(G_{\delta ,N}(w)=\int _\delta ^w|H^{\prime }_{\delta ,N}(t)|^2dt,\ w\ge \delta .\) 
Then for \(w\ge \delta \) ,
\(H(w)\le w^{\beta },\) 
\( wH^{\prime }(w)\le \beta w^{\beta },\) 
and
\(G(w)\le \frac{1}{2k_1-1}wG^{\prime }(w).\) 
Here and in the sequel we omit the subscripts in \(G_{\delta ,N}\)  and \(H_{\delta ,N}\) .
Let \(\delta =R(\left\Vert \mathbf {b}^s\right\Vert _{L^{\infty }}+\left\Vert \mathbf {b}^a\right\Vert _{\operatorname{BMO}})\) , and define \(\Psi =\left|\widetilde{\varphi }-c_0\right|+\delta \) , where \(c_0\)  is an arbitrary constant. Then \(\Psi \)  is a subsolution to the equation \(\operatorname{div}_x(A_{||}\nabla _x\widetilde{\varphi })=\operatorname{div}_x(\mathbf {b}\mathbb {1}_{5Q}-(\mathbf {b}^a)_{2Q})\) . Also, since \(\Psi \ge \delta \) , one can define \(H(\Psi )\) , \(G(\Psi )\)  etc.
For any \(R\le r^{\prime }<r\le 2R\) , let \(\eta \in C_0^2(B_r)\)  with \(\eta =1\)  in \(B_{r^{\prime }}\)  and \(\left|\nabla \eta \right|\lesssim (r-r^{\prime })^{-1}\) . Choose \(v=G(\Psi )\eta ^2>0\)  as a test function. Then since \(\Psi \)  is a subsolution, one has
\(\int _{B_r}A_{||}\nabla \Psi \cdot \nabla v\le \int _{B_r}\mathbf {b}\cdot \nabla v.\) 
For the left-hand side of (REF ), we have (see the proof of Lemma 3.4 of [1]})
\(\int _{B_r}A_{||}^s\nabla \Psi \cdot \nabla v\ge \frac{\lambda _0}{2}\int _{B_r}\left|\nabla H(\Psi )\right|^2\eta ^2-\frac{C(n,\lambda _0,k_0)}{(2k_0-1)^2}\frac{\beta ^2r^n}{(r-r^{\prime })^2}\left(_{B_r}\Psi ^{\beta q}\right)^{2/q},\) 
and
\(\int _{B_r}A_{||}^a\nabla \Psi \cdot \nabla v\le \frac{\lambda _0}{8}\int _{B_r}\left|\nabla H(\Psi )\right|^2\eta ^2\\+\frac{C(n,\lambda _0,\Lambda _0,q,k_0)}{(2k_0-1)^2}\frac{\beta ^2r^n}{(r-r^{\prime })^2}\left(_{B_r}\Psi ^{\beta q}\right)^{2/q}.\) 
The right-hand side of (REF ) equals
\(\int _{B_r}\mathbf {b}^s\cdot \nabla \left(G(\Psi )\eta ^2\right)+\int _{B_r}\mathbf {b}^a\cdot \nabla \left(G(\Psi )\eta ^2\right)\\=\int _{B_r}\mathbf {b}^s\cdot \nabla H(\Psi )\left|H^{\prime }(\Psi )\right|\eta ^2+2\int _{B_r}\mathbf {b}^s\cdot \nabla \eta G(\Psi )\eta \\+\int _{B_r}\left(\mathbf {b}^a-(\mathbf {b}^a)_{B_{r}}\right)\cdot \nabla H(\Psi )\left|H^{\prime }(\Psi )\right|\eta ^2+2\int _{B_r}\left(\mathbf {b}^a-(\mathbf {b}^a)_{B_{r}}\right)\cdot \nabla \eta G(\Psi )\eta \\=: I_1+I_2+I_3+I_4.\) 
Using Cauchy-Schwartz inequality, (REF ), as well as Young's inequality, we obtain
\(\left|I_1\right|\le \frac{\lambda _0}{8}\int \left|\nabla H(\Psi )\right|^2\eta ^2+C(n,\lambda _0)\left\Vert \mathbf {b}^s\right\Vert _{L^{\infty }}^2\beta ^2\int \Psi ^{2\beta -2}\eta ^2.\) 

Recall, in addition, that \(\Psi \ge \delta =R(\left\Vert \mathbf {b}^s\right\Vert _{L^{\infty }}+\left\Vert \mathbf {b}^a\right\Vert _{\operatorname{BMO}})\)  and \(2<q<\frac{2n}{n-2}\) . Then \(\left|I_1\right|\)  is bounded by
\(\frac{\lambda _0}{8}\int \left|\nabla H(\Psi )\right|^2\eta ^2+C(n,\lambda _0)\beta ^2R^{-2}\int \Psi ^{2\beta }\eta ^2\\\le \frac{\lambda _0}{8}\int \left|\nabla H(\Psi )\right|^2\eta ^2+C(n,\lambda _0)\beta ^2R^{-2}r^n\left(_{B_r}\Psi ^{q\beta }\right)^{2/q}.\) 
For \(I_2\) , we use (REF ) and obtain
\(\left|I_2\right|&\le \frac{\left\Vert \mathbf {b}^s\right\Vert _{L^{\infty }}}{r-r^{\prime }}\frac{\beta ^2}{2k_1-1}\int \Psi ^{2\beta -1}\left|\eta \right|\le \frac{C(n,k_0)\beta ^2}{(r-r^{\prime })(2k_0-1)R}\int _{B_r}\Psi ^{2\beta }\nonumber \\&\le \frac{C(n,k_0)\beta ^2r^n}{(r-r^{\prime })(2k_0-1)R}\left(_{B_r}\Psi ^{q\beta }\right)^{2/q}.\) 

Turning to \(I_3\) , we estimate
\(\left|I_3\right|&\le \left(\int _{B_r}\left|\mathbf {b}^a-(\mathbf {b}^a)_{B_r}\right|^{s_0^{\prime }}\right)^{\frac{1}{s_0^{\prime }}}\left(\int \left|\nabla H(\Psi )\right|^2\eta ^2\right)^{1/2}\left(\int \left|H^{\prime }(\Psi )\right|^q\eta ^q\right)^{1/q}\nonumber \\&\le C(n,q)\left\Vert \mathbf {b}^a\right\Vert _{\operatorname{BMO}}r^{\frac{n}{s_0^{\prime }}}\left(\int \left|\nabla H(\Psi )\right|^2\eta ^2\right)^{1/2}\beta \left(\int \Psi ^{q\beta -q}\eta ^q\right)^{1/q}\nonumber \\&\le \frac{\lambda _0}{8}\int \left|\nabla H(\Psi )\right|^2\eta ^2+\frac{C(n,\lambda _0,q)\beta ^2r^n}{R^2}\left(_{B_r}\Psi ^{q\beta }\right)^{2/q},\) 

where \(s_0^{\prime }=\frac{s_0}{s_0-1}\) .
Finally, for \(I_4\) , we have
\(\left|I_4\right|&\le \left(\int _{B_r}\left|\mathbf {b}^a-(\mathbf {b}^a)_{B_r}\right|^{\frac{q}{q-2}}\right)^{\frac{q-2}{q}}\left(\int \left|G(\Psi )\right|^{q/2}\left|\nabla \eta \right|^{q/2}\right)^{2/q}\nonumber \\&\le C(n,q)\left\Vert \mathbf {b}^a\right\Vert _{\operatorname{BMO}}r^{\frac{(q-2)n}{q}}\frac{1}{2k_1-1}\frac{\beta ^2}{r-r^{\prime }}\left(\int _{B_r}\Psi ^{q\beta -\frac{q}{2}}\right)^{2/q}\nonumber \\&\le C(n,q,k_0)\frac{\beta ^2r^n}{(2k_0-1)(r-r^{\prime })R}\left(_{B_r}\Psi ^{q\beta }\right)^{2/q}.\) 
Combining (REF )–(REF ), we get
\(\frac{\lambda _0}{8}_{B_{r^{\prime }}}\left|\nabla H(\Psi )\right|^2\le C\beta ^2\left((r-r^{\prime })^{-2}+(r-r^{\prime })^{-1}R^{-1}+R^{-2}\right)\left(_{B_r}\Psi ^{q\beta }\right)^{2/q}.\) 
Furthermore, since \(\alpha =2\)  when \(n\ge 3\)  and \(\alpha \in (1,2)\)  when \(n=2\) , by
Sobolev embedding
\(\left(_{B_{r^{\prime }}}H(\Psi )^{\frac{n\alpha }{n-\alpha }}\right)^{\frac{n-\alpha }{n\alpha }}\lesssim \left(_{B_{r^{\prime }}}H(\Psi )^2\right)^{\frac{1}{2}}+r^{\prime }\left(_{B_{r^{\prime }}}\left|\nabla H(\Psi )\right|^2\right)^{\frac{1}{2}}.\) 
Now by (REF ), (REF ), and letting \(N\)  go to infinity, we obtain
\(\left(_{B_{r^{\prime }}}\Psi ^{\beta \frac{n\alpha }{n-\alpha }}\right)^{\frac{n-\alpha }{n\alpha }}\le \left(_{B_{r^{\prime }}}\Psi ^{2\beta }\right)^{1/2}\\+ C\beta r^{\prime }\left((r-r^{\prime })^{-2}+(r-r^{\prime })^{-1}R^{-1}+R^{-2}\right)^{1/2}\left(_{B_r}\Psi ^{q\beta }\right)^{1/q}\\\le C\left(1+\beta \left(\frac{r^{\prime }}{r-r^{\prime }}+\frac{r^{\prime }}{\sqrt{(r-r^{\prime })R}}+\frac{r^{\prime }}{R}\right)\right)\left(_{B_r}\Psi ^{q\beta }\right)^{1/q}.\) 
Letting \(l=\frac{n\alpha }{(n-\alpha )q}>1\) , \(\beta =\beta _i=kl^i\) , \(r=r_i=R+\frac{R}{2^i}\)  and \(r^{\prime }=r_{i+1}\)  for \(i=0,1,2,\dots \) , one finds
\(\left(_{B_{r_{i+1}}}\Psi ^{kl^{i+1}q}\right)^{\frac{1}{kl^{i+1}q}}&\le (Ckl^i)^{\frac{1}{kl^i}}\left(_{B_{r_i}}\Psi ^{kl^iq}\right)^{\frac{1}{kl^iq}}\le \dots \\&\le (Ck)^{\sum _{j=0}^i\frac{1}{kl^j}}l^{\sum _{j=0}^i\frac{j}{kl^j}}\left(_{B_{2R}}\Psi ^{kq}\right)^{\frac{1}{kq}}.\) 

Letting \(i\rightarrow \infty \) , we have \(\sup _{B_R}\Psi \le C(n,\lambda _0,\Lambda _0,q,k_0)\left(_{B_{2R}}\Psi ^{kq}\right)^{\frac{1}{kq}}\) , and thus
\(\sup _{B_R}\left|\widetilde{\varphi }-c_0\right|\le C\left(_{B_{2R}}\left|\widetilde{\varphi }-c_0\right|^{kq}\right)^{\frac{1}{kq}}+ C\delta ,\) 
where \(C=C(n,\lambda _0,\Lambda _0,q,k_0)\) . Choosing \(k\)  and \(q\)  such that \(kq=p\)  yields (REF ). The proof of (REF ) is similar and thus omitted.
